Structure of attractors for boundary maps associated to Fuchsian groups (Q1681817)

Let \(\mathbb{S}\) denote the Euclidean boundary of the upper half-plane model or the unit disk model of the hyperbolic plane. For a finitely generated Fuchsian group, \textit{R. Bowen} and \textit{C. Series} [Publ. Math., Inst. Hautes Étud. Sci. 50, 153--170 (1979; Zbl 0439.30033)] studied a boundary map from \(\mathbb{S}\) to itself which has the Markov property with respect to a partition of \(\mathbb{S}\). In this paper the authors consider generalized boundary maps with respect to a set of jumps for compact torsion-free Fuchsian groups and study the dynamical properties of the maps. As a natural extension of the boundary map \(f\), a two-dimensional map \(F\) is defined from \(\mathbb{S}\times \mathbb{S}\setminus \Delta\) to itself, where \(\Delta\) denotes the diagonal of \(\mathbb{S}\times \mathbb{S}\). Let \(\bar{A}=\{A_1,\ldots, A_{8g-4}\}\) be a set of partition points on \(\mathbb{S}\), where these points are taken from the intervals associated with a \((8g-4)\)-sided fundamental domain of the Fuchsian group. Then the boundary map \(f_{\bar{A}}\) and the corresponding two-dimensional map \(F_{\bar{A}}\) are defined by the \(8g-4\) side-pairing maps. The authors show that if each partition point \(A_i\) satisfies the short cycle property, then there exists a bijectivity domain of \(F_{\bar{A}}\) in \(\mathbb{S}\times\mathbb{S}\setminus \Delta\) such that it has a finite rectangular structure and is a global attractor for \(F_{\bar{A}}\).